CN110032073B - 1/2 power attraction repetitive control method with equivalent disturbance compensation - Google Patents

1/2 power attraction repetitive control method with equivalent disturbance compensation Download PDF

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CN110032073B
CN110032073B CN201910386927.6A CN201910386927A CN110032073B CN 110032073 B CN110032073 B CN 110032073B CN 201910386927 A CN201910386927 A CN 201910386927A CN 110032073 B CN110032073 B CN 110032073B
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error
attraction
equivalent disturbance
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孙明轩
李旺
王晗
邹胜祥
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Zhejiang University of Technology ZJUT
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Abstract

A1/2 power attraction repetitive control method with equivalent disturbance compensation is disclosed, wherein a given module generates periodic reference signals, a periodic feedforward link is constructed, equivalent disturbance compensation is introduced into the 1/2 power attraction law, and a disturbance observer is used for estimating equivalent disturbance; constructing an ideal error dynamic state based on the power attraction law, designing a controller according to the ideal error dynamic state, and taking a signal obtained by calculation as the control input of a servo system; the specific controller parameter setting can be carried out according to the convergence performance index of the representation system, and a calculation formula for representing a monotone decreasing area, an absolute attraction layer, a steady-state error band boundary and the maximum step number of the tracking error entering the steady-state error band for the first time in the tracking error convergence process is provided. The 1/2 power attraction repetitive control method with equivalent disturbance compensation provided by the invention can improve the tracking precision of the system and completely inhibit periodic disturbance by estimating the equivalent disturbance.

Description

1/2 power attraction repetitive control method with equivalent disturbance compensation
Technical Field
The invention designs a 1/2 power attraction repetitive control method with equivalent disturbance compensation, which is used for a periodic position servo system and is also suitable for other industrial occasions containing periodic operation processes.
Background
The essence of the internal model principle is that a dynamic model of external signals of the system (namely, the internal model) is implanted into the control system, so that a high-precision feedback control system is formed, and the system can track input signals without errors. Repetitive control is a typical application of the principle of the internal model, and regardless of the specific form of the input signal, as long as the initial segment signal is given, the internal model accumulates the input signal cycle by cycle, and repeatedly outputs the same signal as the previous cycle. The repetitive controller designed according to the internal model principle has the characteristics of memory and learning, and realizes the track tracking of periodic reference signals and the complete suppression of periodic disturbance. At present, the repetitive control technology has been successfully applied to the precise control of servo motors, the power electronic control technology, the power quality control and the like.
Different from a method of designing dynamics based on a sliding mode function by an approach law, the approach law is a method of designing dynamics based on errors directly, and the approach law method can enable the errors to be converged in a limited time. The attraction law method directly utilizes the tracking error signal, and the controller design is more direct and concise. The attraction law can reflect the expected dynamic characteristics of the system error when the disturbance condition is not considered; under the condition of interference, the controller cannot be directly designed according to the attraction law, so that interference suppression measures are 'embedded' into the attraction law, an ideal error dynamic state with a disturbance suppression effect is constructed, and the digital controller is designed according to the constructed ideal error dynamic equation. The closed loop system dynamics is determined by the ideal error dynamics and has a desired tracking performance characterized by the ideal error dynamics. In the attraction law method, the performance of the system is characterized by a monotone decreasing area, an absolute attraction layer, a steady-state error band and the maximum number of steps required for the tracking error to enter the steady-state error band for the first time. Once an ideal error dynamic form is given, specific expressions of various indexes are given in advance and used for guiding the parameter setting of the controller.
The extended state observer is a core unit for active disturbance rejection control, and the basic method is to define the total disturbance (including internal disturbance and external disturbance) as a new state, construct a disturbance observer of an extended variable (including total disturbance action) by using a method designed by the extended state observer. The method can estimate system variables, and can also estimate the real-time action quantity of overall disturbance in a system model, so as to compensate the influence of disturbance signals. Since the overall disturbance encompasses uncertainties in the system model, the system model is greatly simplified, and the control gain is considered known, facilitating controller design.
Disclosure of Invention
In order to solve the problem that the existing power attraction repetitive controller can not realize high-precision tracking of a servo system and can not completely inhibit periodic disturbance, the invention provides a 1/2 power attraction repetitive control method with equivalent disturbance compensation, in order to enable a closed-loop system to have preset expected error tracking performance, a motor servo repetitive controller is designed according to an ideal error dynamic equation of a power attraction structure, non-periodic components exist in disturbance while the periodic disturbance components are completely inhibited, a disturbance observer is introduced into the closed-loop system to compensate non-periodic disturbance, the control performance is further improved, and the motor servo system can realize high-speed and high-precision tracking; the invention expands the disturbance effect influencing the system output into a new variable to construct a disturbance observer, the disturbance observer does not need to directly measure a disturbance signal and know a specific model of the disturbance signal, and the invention specifically provides a specific expression of at most four indexes, namely a steady state error band, an absolute attraction layer, a monotone subtraction area and a step number required for a tracking error to enter the steady state error band for the first time, and the specific expression is used for guiding the parameter setting of the controller.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a1/2 power attraction repetitive control method with equivalent disturbance compensation comprises the following steps:
step 1, giving periodic reference signals to satisfy
rk＝rk-N (1)
Where N is the period of the reference signal, rkAnd rk-NReference signals respectively representing time k and time k-N;
step 2, defining tracking error
Figure BDA0002055138910000021
In the formula
Figure BDA0002055138910000022
Figure BDA0002055138910000023
Figure BDA0002055138910000024
Satisfy the requirement of
Figure BDA0002055138910000025
Wherein e isk+1Represents the tracking error at time k +1, rk+1Reference signal, y, representing the time instant k +1k+1、yk、 yk+1-NAnd yk-NRepresenting the output signals at times k +1, k +1-N and k-N, respectively, ukAnd uk-NRepresenting the input signal at times k and k-N, wk+1And wk+1-NRespectively representing the interfering signals at times k +1 and k +1-N, d represents the delay, A (q)-1) And B (q)-1) Is q-1Polynomial of (a), q-1Representing a one-step delay operator, naRepresents A (q)-1) Order of (1), nbRepresents B (q)-1) The order of (a) is selected,
Figure BDA0002055138910000026
is a system parameter and b0≠0， na≥nbD is an integer, and d is not less than 1;
step 3. constructing equivalent disturbance
dk＝wk-wk-N (4)
Where N is the period of the reference signal, dkRepresenting the equivalent disturbance signal at time k, wkAnd wk-NRespectively representing interference signals at the k moment and the k-N moment;
expressing the tracking error as
ek+1＝rk+1-yk+1-N+A1(q-1)(yk-yk-N)-q-d+1B(q-1)(uk-uk-N)-dk+1 (5)
Wherein d isk+1Representing the equivalent disturbance at the moment k + 1;
step 4, designing an observer, estimating equivalent disturbance, and carrying out the following process:
design observer equivalent disturbance dk+1Observing, and compensating equivalent disturbance by the observed value, wherein two observed variables of the observer are
Figure BDA0002055138910000031
And
Figure BDA0002055138910000032
estimate e separatelykAnd dkBased on the error dynamics (equation (5)), an observer of the following form is designed
Figure BDA0002055138910000033
Wherein the content of the first and second substances,
Figure BDA0002055138910000034
represents the error ek+1Is estimated by the estimation of (a) a,
Figure BDA0002055138910000035
represents the error ekIs estimated by the estimation of (a) a,
Figure BDA0002055138910000036
representing equivalent perturbation, beta1Representing the observer gain coefficient, beta, with respect to the error2Representing the observer gain coefficient with respect to the equivalent disturbance,
Figure BDA0002055138910000037
an estimation error representing a tracking error;
estimation error of equivalent disturbance
Figure BDA0002055138910000038
Is composed of
Figure BDA0002055138910000039
Estimation error of tracking error is
Figure BDA00020551389100000310
The expressions (7) and (8) are written as follows
Figure BDA00020551389100000311
Note the book
Figure BDA0002055138910000041
The characteristic equation is
|λI-B|＝0 (10)
Namely, it is
λ2+(β1-β2-1)λ-β1＝0 (11)
Thus, the characteristic root is
Figure BDA0002055138910000042
For parameter beta1And beta2Configured so that all feature roots are within the unit circle, matrix B is a Schur stable matrix, and the estimation error converges asymptotically, i.e.
Figure BDA0002055138910000043
Step 5. construct the 1/2 power attraction law with disturbance suppression measures
Figure BDA0002055138910000044
Wherein rho and epsilon are both adjustable parameters, rho is more than 0 and less than 1, and epsilon is more than 0;
and 6, constructing a repetitive controller with equivalent disturbance compensation, wherein the process is as follows:
combining equation (5) and equation (12), design a repetitive controller with equivalent disturbance compensation
Figure BDA0002055138910000045
Note the book
Figure BDA0002055138910000046
Expressing a repetitive controller as
uk＝uk-N+vk (14)
Will ukAs input signal of controller of servo object, measuring to obtain output signal y of servo systemkFollows the reference signal rkAnd (4) changing.
Further, an expression of four indexes, such as a steady state error band, an absolute attraction layer, a monotone decreasing area, the maximum number of steps required for the tracking error to enter the steady state error band for the first time and the like is given, and the expression is used for describing the tracking performance of the system and guiding the parameter setting of the controller, wherein the steady state error band, the absolute attraction layer, the monotone decreasing area and the maximum convergence number are defined as follows:
1) monotonous decreasing region deltaMDR: when e iskGreater than this boundary, ekThe same number is decreased, namely the following conditions are met:
Figure BDA0002055138910000051
2) absolute attraction layer ΔAAL: absolute value of system tracking errorkIf | is greater than this boundary, its | ekI, monotonically decreases, i.e. the condition is satisfied:
Figure BDA0002055138910000052
3) steady state error band ΔSSE: when the system error once converges into the boundary, the error is stabilized in the region, that is, the following condition is satisfied:
Figure BDA0002055138910000053
4) maximum number of convergence steps
Figure BDA0002055138910000054
The tracking error passes through at most
Figure BDA0002055138910000055
Entering a steady state error band;
equivalent disturbance compensation error satisfaction
Figure BDA0002055138910000056
Specific expressions of the respective indices are as follows
1) Monotonous decreasing region deltaMDR
Figure BDA0002055138910000057
2) Absolute attraction layer ΔAAL
Figure BDA0002055138910000058
3) Steady state error band ΔSSE
Figure BDA0002055138910000061
Wherein the content of the first and second substances,
Figure BDA0002055138910000062
4) number of convergence steps
Figure BDA0002055138910000063
Figure BDA0002055138910000064
Wherein e is0In order to be the initial value of the tracking error,
Figure BDA0002055138910000065
represents the smallest integer no less than.
The technical conception of the invention is as follows: a power-of-1/2 attraction repetitive controller with equivalent disturbance compensation is provided. According to a given reference signal and the constructed equivalent disturbance, an observer is introduced to estimate the equivalent disturbance, and interference suppression measures are embedded into a power law of attraction to form ideal error dynamics with interference suppression effect, so that a repetitive controller with equivalent disturbance compensation is designed, and rapid and high-precision tracking of the given reference signal is realized.
The invention has the following beneficial effects: the method has equivalent disturbance compensation, complete suppression of periodic disturbance, fast convergence performance and high tracking precision.
Drawings
Fig. 1 is a block diagram of a motor apparatus.
FIG. 2 is a block diagram of an equivalent disturbance observer.
Fig. 3 is a block diagram of a power-suction repetition controller.
FIG. 4 is a graph of the time when a disturbance w occursk＝5sin(2πfkTs) +0.15sgn (sin (2k pi/150)), the simulation results for the controller parameters with ∈ 0.1, ρ 0.3, and Δ 0.3, where Δ is plottedMDR，ΔAALAnd deltaSSE。
FIG. 5 is a graph of the time when a disturbance w occursk＝-10sin(2πfkTs) +0.15sgn (sin (2k pi/150)), the simulation results for the controller parameters with ∈ 0.1, ρ 0.3, and Δ 0.3, where Δ is plottedMDR，ΔAALAnd deltaSSE。
FIG. 6 is a graph of the time when a disturbance w occursk＝5sin(2πfkTs) +0.15sgn (sin (2k pi/150)), the simulation results for the controller parameters with ∈ 0.2, ρ 0.3, and Δ 0.3, where Δ is plottedMDR，ΔAALAnd deltaSSE。
FIG. 7 is a graph of the time when a disturbance w occursk＝-10sin(2πfkTs) +0.15sgn (sin (2k pi/150)), where the controller parameter is 0.2, 0.3, and 0.3In the figure,. DELTA.MDR，ΔAALAnd deltaSSE。
FIG. 8 is a graph of the time when a disturbance w occursk＝5sin(2πfkTs) The simulation results for the case of +0.15sgn (sin (2k pi/150)), where the controller parameter ∈ is 0.5, ρ is 0.3, and Δ is 0.5 are shown in the figure, where Δ is indicatedMDR，ΔAALAnd deltaSSE。
FIG. 9 is a graph of the time when a disturbance w occursk＝-10sin(2πfkTs) The simulation results for the case of +0.15sgn (sin (2k pi/150)), where the controller parameter ∈ is 0.5, ρ is 0.3, and Δ is 0.5 are shown in the figure, where Δ is indicatedMDR，ΔAALAnd deltaSSE。
Fig. 10 to 13 show experimental results of the permanent magnet synchronous motor control device when the repetitive controller parameter is ρ 0.7 and ∈ 0.3, where,
FIG. 10 is a reference position signal and an actual position signal under the influence of a repetitive controller based on the power law of attraction;
FIG. 11 is a controller voltage signal under the influence of a repetitive controller based on the power law of attraction;
FIG. 12 is a graph of position error under the influence of a repetitive controller based on the power law of attraction;
fig. 13 is a histogram of the distribution of position errors under the action of a repetitive controller based on the power law of attraction.
Fig. 14-17 show that the repetitive controller parameter is ρ 0.7, and ∈ 0.3, and the disturbance observer parameter is β1＝0.2，β20.5, the experimental results of the permanent magnet synchronous motor control device, wherein,
FIG. 14 is a reference position signal and an actual position signal under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation;
FIG. 15 is a controller voltage signal under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation;
FIG. 16 is a position error under repetitive controller action based on the power law of attraction and equivalent disturbance compensation;
fig. 17 is a histogram of the position error distribution under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation.
Fig. 18 to 21 show experimental results of the permanent magnet synchronous motor control device when the repetitive controller parameter is ρ 0.5 and ∈ 0.15, where,
FIG. 18 is a graph of a reference position signal and an actual position signal under the influence of a repetitive controller based on the power law of attraction;
FIG. 19 is a controller voltage signal under the influence of a repetitive controller based on the power law of attraction;
FIG. 20 is a graph of position error under the influence of a repetitive controller based on the power law of attraction;
fig. 21 is a histogram of a position error distribution under the action of a repetitive controller based on the power law of attraction.
Fig. 22-25 show that the repetitive controller parameter is ρ 0.5, and ∈ 0.15, and the disturbance observer parameter is β1＝0.2，β20.5, the experimental results of the permanent magnet synchronous motor control device, wherein,
FIG. 22 is a graph of the reference position signal and the actual position signal under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation;
FIG. 23 is a controller voltage signal under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation;
FIG. 24 is a graph of position error under repetitive controller action based on the power law of attraction and equivalent disturbance compensation;
fig. 25 is a histogram of the position error distribution under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation.
Fig. 26 to 29 are experimental results of the permanent magnet synchronous motor control device in which p is 0.3 and e is 0.1, respectively, in the repetitive controller parameters,
FIG. 26 is a graph of a reference position signal and an actual position signal under the influence of a repetitive controller based on the power law of attraction;
FIG. 27 is a controller voltage signal under the influence of a repetitive controller based on the power law of attraction;
FIG. 28 is a graph of position error under the influence of a repetitive controller based on the power law of attraction;
fig. 29 is a histogram of the distribution of position errors under the action of a repetitive controller based on the power law of attraction.
In fig. 30-33, the repetitive controller parameter is ρ 0.3, and ∈ 0.1, and the disturbance observer parameter is β1＝0.2，β20.5, the experimental results of the permanent magnet synchronous motor control system, wherein,
FIG. 30 is a graph of a reference position signal and an actual position signal under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation;
FIG. 31 is a controller voltage signal under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation;
FIG. 32 is a graph of position error under repetitive controller action based on the power law of attraction and equivalent disturbance compensation;
fig. 33 is a histogram of the position error distribution under the influence of a repetitive controller based on the power law of attraction and equivalent disturbance compensation.
Fig. 34 to 37 are experimental results of the permanent magnet synchronous motor control device in which p is 0.7 and e is 0.3, where,
FIG. 34 is a graph of a reference position signal and an actual position signal under the action of a feedback controller based on the power law of attraction;
FIG. 35 is a controller voltage signal under the influence of a feedback controller based on the power law of attraction;
FIG. 36 is a graph showing a position error under the action of a feedback controller based on the power law of attraction;
fig. 37 is a histogram of the distribution of position errors under the action of a feedback controller based on the power law of attraction.
Fig. 38 to 41 show that the feedback controller parameter is ρ 0.7, and ∈ 0.3, and the disturbance observer parameter is β1＝0.2，β20.5, the experimental results of the permanent magnet synchronous motor control device, wherein,
FIG. 38 is a reference position signal and an actual position signal under the influence of a feedback controller based on the power law of attraction and equivalent disturbance compensation;
FIG. 39 is a graph of the controller voltage signal under the influence of a feedback controller based on the power law of attraction and equivalent disturbance compensation;
FIG. 40 is a graph of position error under the influence of a feedback controller based on the power law of attraction and equivalent disturbance compensation;
fig. 41 is a position error distribution histogram under the action of a feedback controller based on the power law of attraction and equivalent disturbance compensation.
Fig. 42 to 45 are experimental results of the permanent magnet synchronous motor control device in which p is 0.5 and e is 0.15, where,
FIG. 42 is a graph of a reference position signal and an actual position signal under the action of a feedback controller based on the power law of attraction;
FIG. 43 is a controller voltage signal under the action of a feedback controller based on the power law of attraction;
FIG. 44 is a graph showing a position error under the action of a feedback controller based on the power law of attraction;
fig. 45 is a histogram of the distribution of position errors under the action of a feedback controller based on the power law of attraction.
Fig. 46 to 49 show that the feedback controller parameter is ρ ═ 0.5, and ∈ ═ 0.15, and the disturbance observer parameter is β1＝0.2，β20.5, the experimental results of the permanent magnet synchronous motor control device, wherein,
FIG. 46 is a reference position signal and an actual position signal under the influence of a feedback controller based on the power law of attraction and equivalent disturbance compensation;
FIG. 47 is a controller voltage signal under the influence of a feedback controller based on the power law of attraction and equivalent disturbance compensation;
FIG. 48 is a graph of position error under the influence of a feedback controller based on the power law of attraction and equivalent disturbance compensation;
fig. 49 is a position error distribution histogram under the action of a feedback controller based on the power law of attraction and equivalent disturbance compensation.
Fig. 50 to 53 show experimental results of the permanent magnet synchronous motor control device when the feedback controller parameter is ρ 0.3 and ∈ 0.1, where,
FIG. 50 is a graph of a reference position signal and an actual position signal under the action of a feedback controller based on the power law of attraction;
FIG. 51 is a controller voltage signal under the influence of a feedback controller based on the power law of attraction;
FIG. 52 is a graph showing a position error under the action of a feedback controller based on the power law of attraction;
fig. 53 is a histogram of the distribution of position errors under the action of a feedback controller based on the power law of attraction.
Fig. 54 to 57 show that the feedback controller parameter is ρ ═ 0.3, and ∈ ═ 0.1, and the disturbance observer parameter is β1＝0.2，β20.5, the experimental results of the permanent magnet synchronous motor control device, wherein,
FIG. 54 is a graph of the reference position signal and the actual position signal under the influence of a feedback controller based on the power law of attraction and equivalent disturbance compensation;
FIG. 55 is a graph of the controller voltage signal under the influence of a feedback controller based on the power law of attraction and equivalent disturbance compensation;
FIG. 56 is a graph of position error under the influence of a feedback controller based on the power law of attraction and equivalent disturbance compensation;
fig. 57 is a position error distribution histogram under the action of a feedback controller based on the power law of attraction and equivalent disturbance compensation.
Detailed Description
The following further describes embodiments of the present invention with reference to the accompanying drawings.
Referring to fig. 1-57, a power-of-1/2 attraction repetitive control method with equivalent disturbance compensation, wherein fig. 1 is a block diagram of a motor apparatus; FIG. 2 is a block diagram of an equivalent disturbance observer; fig. 3 is a block diagram of a power attraction repeat controller.
The power attraction repetitive servo control method based on equivalent disturbance compensation comprises the following steps:
step 1, giving a periodic reference signal, and satisfying the requirement (1);
step 2, defining a tracking error, wherein the tracking error of the system is (2);
step 3, constructing an equivalent disturbance (4), and expressing a system tracking error as (5) by using the equivalent disturbance (4);
step 4, designing an observer and estimating equivalent disturbance;
step 5, constructing a 1/2 power attraction law (12) with disturbance suppression measures;
and 6, constructing a repetitive controller with equivalent disturbance compensation, combining the formula (5) and the formula (12), designing a repetitive controller (13) with equivalent disturbance compensation, and expressing the repetitive controller as (14).
The above repetitive controller design is explained as follows:
1) introduction of d into power attraction lawk+1Reflecting suppression measures for disturbing signals of a given periodic pattern, introduced
Figure BDA0002055138910000101
An estimate of the equivalent disturbance is reflected, thereby providing equivalent disturbance compensation.
2) In the formula (13), ek、yk、yk-1、yk-1-NAre all obtained by measurement, uk-1、uk-1-NThe stored value for the control signal is read from memory.
3) When the reference signal satisfies rk＝rk-1The discrete repetitive controller is also suitable for the constant value regulation problem, and the equivalent disturbance is dk＝wk-wk-1(ii) a Wherein r isk-1Reference signal representing the time instant k-1, wk-1Representing the interference signal at time k-1; the feedback controller with equivalent disturbance compensation is
Figure BDA0002055138910000111
4) The discrete time controller is designed for a second-order system, and a design result of a high-order system can be given according to the same method.
Further, a specific expression of four indexes, namely a steady-state error band, an absolute attraction layer, a monotone decreasing area and the maximum step number required for the tracking error to enter the steady-state error band for the first time is given, and the specific expression is used for describing the tracking performance of the system and guiding the parameter setting of the controller.
In this embodiment, for example, the permanent magnet synchronous motor device executes a repetitive tracking task in a fixed interval, and the position reference signal has a periodic symmetry characteristic. TMS320F2812DSP is used as a controller, a Korean LS AC servo motor APM-SB01AGN is used as a control object, and a permanent magnet synchronous motor servo system is formed by the ELMO AC servo driver and an upper mechanism to control the position of the motor. The servo system adopts three-loop control, the current loop and speed loop controller ELMO driver provide, and the position loop is provided by DSP development board.
Obtaining a mathematical model of the servo object by parameter estimation
yk+1-1.8949yk+0.8949yk-1＝1.7908uk-0.5704uk-1+wk+1 (22)
Wherein, yk，ukPosition output and control input, w, respectively, of the position servo systemkIs an interference signal.
The effectiveness of the repetitive controller given by the present invention will be illustrated in this example by numerical simulation and experimental results.
Numerical simulation: in the embodiment, a sinusoidal signal is used as a system reference signal, and the corresponding repetitive controller expression is
Figure BDA0002055138910000112
Given a position reference signal of rk＝20(sin(2πfkTs-1/2 pi) +1) in degrees (deg), frequency f 1Hz, sampling period TsThe number of sampling points N is 1000, 0.005 s. Selecting proper disturbance amount w during simulationkIt consists of periodic disturbances and non-periodic random disturbances.
Under the action of a repetitive controller (23), different controller parameters rho and epsilon are selected, and three boundary layers of a servo system are different. For purposes of illustrating the invention patent with respect to the monotonically decreasing region ΔMDRAbsolute attraction layer DeltaAALAnd steady state error band ΔSSEThe theoretical correctness of (1).
1) When the controller parameter epsilon is 0.1, rho is 0.3, delta is 0.3,
ΔSSE＝ΔAAL＝ΔMDR＝0.7176
2) when the controller parameter epsilon is 0.2, rho is 0.3, delta is 0.3,
ΔSSE＝ΔAAL＝0.5195，ΔMDR＝0.6608
3) when the controller parameter epsilon is 0.5, rho is 0.3, delta is 0.5,
ΔSSE＝0.5893，ΔAAL＝0.5026，ΔMDR＝1.6248
the simulation results are shown in FIGS. 4-9, where FIGS. 4, 6 and 8 are perturbation amounts wk＝5sin(2πfkTs) The simulation result of +0.15sgn (2k pi/150), FIGS. 5, 7 and 9 are disturbance amounts wk＝-10sin(2πfkTs) +0.15sgn (2k pi/150).
The numerical results verify the monotonous reduction area delta of the tracking error of the system under the action of the repetitive controller given by the patent under the condition of a given system model, a reference signal and an interference signalMDRAbsolute attraction layer DeltaAALAnd steady state error band ΔSSEThe accuracy of (2).
Experimental verification the block diagram of the motor apparatus used in the experiment is shown in figure 1. The tracking performance of the discrete controller based on the 1/2 power attraction law is verified by setting different controller parameters. Given position signal rkA (sin (2 pi × (k-200)/N) +1), where the amplitude a is 135deg, and the sampling period Ts2.5ms, k is the number of samples, and N is 800.
The adopted repetitive controller has the following form
Figure BDA0002055138910000121
The repetitive controller based on disturbance compensation is adopted and has the following form
Figure BDA0002055138910000122
The feedback controller adopted has the following form
Figure BDA0002055138910000123
The repetitive controller based on disturbance compensation is adopted and has the following form
Figure BDA0002055138910000131
1) The controller (24) is adopted, the controller parameters are rho is 0.7, epsilon is 0.3, and the experimental results are shown in figures 10-13, wherein delta is shown in the figuresSSE＝0.1deg。
2) A controller (25) is adopted, wherein rho and epsilon are taken as parameters of the controller and 0.7 and 0.3 respectively, and beta is taken as a parameter of an equivalent disturbance observer1＝0.2，β2The results are shown in FIGS. 14-17, where Δ is 0.5SSE＝0.08deg。
3) The controller (24) is adopted, the controller parameters are rho is 0.5, epsilon is 0.15, and the experimental results are shown in figures 18-21, wherein delta is shown in the figuresSSE＝0.14deg。
4) A controller (25) is adopted, wherein rho and epsilon are respectively taken as parameters of the controller and 0.5 and 0.15, and beta is taken as a parameter of an equivalent disturbance observer1＝0.2，β2The results are shown in FIGS. 22-25, where Δ is 0.5SSE＝0.11deg。
5) The controller (24) is adopted, the controller parameters are rho is 0.3, epsilon is 0.1, and the experimental results are shown in figures 26-29, wherein delta is shown in the figuresSSE＝0.1deg。
6) A controller (25) is adopted, wherein rho and epsilon are respectively taken as parameters of the controller and 0.3 and 0.1, and beta is taken as a parameter of an equivalent disturbance observer1＝0.2，β2The results are shown in FIGS. 30-33, where Δ is 0.5SSE＝0.07deg。
7) The controller (26) is adopted, the controller parameters are rho is 0.7, epsilon is 0.3, and the experimental results are shown in FIGS. 34-37, wherein delta is shown in the figureSSE＝0.13deg。
8) A controller (27) is adopted, wherein rho and epsilon are respectively taken as the parameters of the controller and 0.7 and 0.3, and beta is taken as the parameter of an equivalent disturbance observer1＝0.2，β2The results are shown in FIGS. 38-41, where Δ is 0.5SSE＝0.11deg。
9) The controller (26) is adopted, the controller parameters are rho is 0.5, epsilon is 0.15, and the experimental results are shown in figures 42-45, wherein delta is shown in the figuresSSE＝0.16deg。
10) A controller (27) is adopted, wherein rho and epsilon are respectively taken as the parameters of the controller and 0.5 and 0.15, and beta is taken as the parameter of an equivalent disturbance observer1＝0.2，β2The results are shown in FIGS. 46-49, where Δ is 0.5SSE＝0.13deg。
11) The controller (26) is adopted, the controller parameters are rho is 0.3, epsilon is 0.1, and the experimental results are shown in figures 50-53, wherein delta is shown in the figuresSSE＝0.13deg。
12) A controller (27) is adopted, wherein rho and epsilon are respectively taken as the parameters of the controller and 0.3 and 0.1, and beta is taken as the parameter of an equivalent disturbance observer1＝0.2，β2The results are shown in FIGS. 54-57, where Δ is 0.5SSE＝0.095deg。
The experiment result shows that the equivalent disturbance is introduced and is estimated by the equivalent disturbance observer, the compensation for the unmodeled characteristic and the external unknown disturbance of the system is provided, and the influence of the unknown disturbance on the tracking performance can be effectively inhibited; and the periodic disturbance is completely inhibited by adopting repeated control, so that the control performance of the system is further improved.

Claims (2)

1. A1/2 power attraction repetitive control method with equivalent disturbance compensation, wherein a controlled object is a periodic servo system, and the method is characterized by comprising the following steps:
step 1, giving periodic reference signals to satisfy
rk＝rk-N (1)
Where N is the period of the reference signal, rkAnd rk-NReference signals respectively representing time k and time k-N;
step 2, defining tracking error
Figure FDA0003465186540000011
In the formula
Figure FDA0003465186540000012
Figure FDA0003465186540000013
Figure FDA0003465186540000014
Satisfy the requirement of
A(q-1)yk＝q-dB(q-1)uk+wk (3)
Wherein e isk+1Represents the tracking error at time k +1, rk+1Reference signal, y, representing the time instant k +1k+1、yk、yk+1-NAnd yk-NRepresenting the output signals at times k +1, k +1-N and k-N, respectively, ukAnd uk-NRepresenting the input signal at times k and k-N, wk+1And wk+1-NRespectively representing the interfering signals at times k +1 and k +1-N, d represents the delay, A (q)-1) And B (q)-1) Is q-1Polynomial of (a), q-1Representing a one-step delay operator, naRepresents A (q)-1) Order of (1), nbRepresents B (q)-1) The order of (a) is selected,
Figure FDA0003465186540000015
is a system parameter and b0≠0，na≥nbD is an integer, and d is not less than 1;
step 3. constructing equivalent disturbance
dk＝wk-wk-N (4)
Where N is the period of the reference signal, dkRepresenting the equivalent disturbance signal at time k, wkAnd wk-NRespectively representing interference signals at the k moment and the k-N moment;
expressing the tracking error as
ek+1＝rk+1-yk+1-N+A1(q-1)(yk-yk-N)-q-d+1B(q-1)(uk-uk-N)-dk+1 (5)
Wherein d isk+1Representing the equivalent disturbance at the moment k + 1;
step 4, designing an observer, estimating equivalent disturbance, and carrying out the following process:
design observer equivalent disturbance dk+1Observing and compensating equivalent disturbance by the observed value; two observed variables of the observer are
Figure FDA0003465186540000021
And
Figure FDA0003465186540000022
are used to estimate e respectivelykAnd dk(ii) a From the error dynamics (equation (5)), an observer of the following form is designed
Figure FDA0003465186540000023
Wherein the content of the first and second substances,
Figure FDA0003465186540000024
represents the error ek+1Is estimated by the estimation of (a) a,
Figure FDA0003465186540000025
represents the error ekIs estimated by the estimation of (a) a,
Figure FDA0003465186540000026
representing equivalent perturbation, β1Representing the observer gain coefficient, beta, with respect to the error2Representing observer gain coefficients with respect to equivalent disturbances;
Figure FDA0003465186540000027
show the heelAn estimation error of the tracking error;
estimation error of equivalent disturbance
Figure FDA0003465186540000028
Is composed of
Figure FDA0003465186540000029
Estimation error of tracking error is
Figure FDA00034651865400000210
The expressions (7) and (8) are written as follows
Figure FDA00034651865400000211
Note the book
Figure FDA00034651865400000212
The characteristic equation is
|λI-B|＝0 (10)
Namely, it is
λ2+(β1-β2-1)λ-β1＝0 (11)
Thus, the characteristic root is
Figure FDA0003465186540000031
For beta is1And beta2Is configured such that all feature roots are within the unit circle;
step 5. construct the 1/2 power attraction law with disturbance suppression measures
Figure FDA0003465186540000032
Wherein rho and epsilon are both adjustable parameters, rho is more than 0 and less than 1, and epsilon is more than 0;
and 6, constructing a repetitive controller with equivalent disturbance compensation, wherein the process is as follows:
combining equation (5) and equation (12) to obtain a repetitive controller with equivalent disturbance compensation
Figure FDA0003465186540000033
Note the book
Figure FDA0003465186540000034
Expressing a repetitive controller as
uk＝uk-N+vk (14)
Will ukAs input signal of controller of servo object, measuring to obtain output signal y of servo systemkFollows the reference signal rkAnd (4) changing.
2. The 1/2 power-of-attraction repetitive control method with equivalent disturbance compensation as claimed in claim 1, wherein expressions of four indexes, namely a steady-state error band, an absolute attraction layer, a monotone decreasing region and the maximum number of steps required for a tracking error to firstly enter the steady-state error band, are given for describing the tracking performance of the system and guiding the parameter setting of the controller, wherein the steady-state error band, the absolute attraction layer, the monotone decreasing region and the convergence steps are defined as follows:
1) monotonous decreasing region deltaMDR: when e iskGreater than this boundary, ekThe same number is decreased, namely the following conditions are met:
Figure FDA0003465186540000035
2) absolute attraction layer ΔAAL: absolute value of system tracking errorkIf | is greater than this boundary, its | ekI, monotonically decreases, i.e. the condition is satisfied:
Figure FDA0003465186540000036
3) steady state error band ΔSSE: when the system error once converges into the boundary, the error is stabilized in the region, that is, the following condition is satisfied:
Figure FDA0003465186540000041
4) maximum number of convergence steps
Figure FDA0003465186540000042
The tracking error passes through at most
Figure FDA0003465186540000043
Entering a steady state error band; equivalent disturbance compensation error satisfaction
Figure FDA0003465186540000044
Specific expressions of the respective indices are as follows
1) Monotonous decreasing region deltaMDR
Figure FDA0003465186540000045
2) Absolute attraction layer ΔAAL
Figure FDA0003465186540000046
3) Steady state error band
Figure FDA0003465186540000047
Wherein the content of the first and second substances,
Figure FDA0003465186540000048
4) number of convergence steps
Figure FDA0003465186540000049
Figure FDA0003465186540000051
Wherein e is0In order to be the initial value of the tracking error,
Figure FDA0003465186540000052
represents the smallest integer no less than.
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